On an operator identity central to projection operator methodology

Abstract

Derivations of master equations using projection operator methodology are based upon an identity whose validity has been established in only limited contexts. Its proof requires precise definitions of e Lt and e ( I−P)L( I−P)t , where L is the Liouville operator and P the projection operator associated with the limited system information of interest. Here, for interacting particles confined to a box, the existence and uniqueness of system dynamics is demonstrated. A distributional extension L of L defined in an L 1 space is derived for which e Lt is the corresponding updating operator. Attempts to define e ( I−P) L( I−P)t within current semigroup theory are outlined, and a possible future approach indicated.